Re: RDF Semantics - Intuitive summary needs to be scoped to interpretations (ISSUE-149)

Hi Pat,

I'll try to start with a summary of where I think we stand in this 
discussion, and then follow with in-line responses:

  - I assume that "miracle" processes (such as ostension and natural 
language) will be used to fix the denotations of a *few* foundational 
IRIs, but for scalability and machine processability, most will be 
defined mostly by RDF descriptions, this will inevitably lead to 
underdeterminism.  In contrast, you seem to assume that most IRIs will 
be defined by "miracle" processes to avoid this underdeterminism.  By 
"miracle" processes, I'm referring to processes that we cannot formalize 
or fully define, and I'm using the term "miracle" in reference to this:

  - We seem to agree that when underdeterminism does occur, it can lead 
to inconsistencies as one RDF author makes one assumption and another 
RDF author independently and unknowingly makes a different (and 
conflicting) assumption.

  - We agree that underdeterminism and disagreement both cause 
inconsistency when graphs are merged as is, and may be formally 

  - I believe that inconsistency that is caused by underdeterminism is 
qualitatively different from inconsistency that is caused by 
disagreement, because with underdeterminism the relevant information 
content of the graphs can be retained by merging a transformation of the 
source graphs -- produced by IRI splitting -- whereas with disagreement 
relevant information content must be discarded.  By "relevant 
information content" I'm talking about the information content of the 
graph other than the particular choice of IRIs that were assigned -- a 
kind of graph isomorphism (though more general than the notion of 
isomorphism that is defined in the RDF Semantics, which is only 
isomorphism over bnode relabeling rather than IRI relabeling).  I have 
not yet been able to determine whether you agree with this difference or 
its significance.  Do you?

  - I believe that it is relevant and useful to acknowledge that 
different interpretations can be applied to different graphs, because 
this in essence is what different applications, such as in the toucan 
example, are doing when they process RDF.  AFAICT you seem to view this 
as an irrelevant non-conforming use of the RDF Semantics.  In contrast, 
I view it as in conformance with the RDF Semantics, since the RDF 
Semantics can be applied to *any* interpretation/graph pairs.

  - We agree that RDF's model theoretic semantics can be used to express 
real world truth.

  - We disagree on the relevance of real world truth in RDF.  You seem 
to view it as the end itself; I view it as a partially relevant means to 
an end, where the real end goal is to facilitate **useful machine 
processing**, and that is not tightly dependent on real world truth.

  - I *think* we agree that real world truth is irrelevant to RDF's 
formal semantics though, i.e., there is no requirement that RDF be used 
only to express real world truths, i.e., the question of whether or not 
an RDF expresses real world truth is outside the scope of the RDF Semantics.

  - AFAICT, the main point of disagreement is that I think it is normal 
and natural to talk about what an IRI denotes within a graph (assuming 
that graph is true), whereas you seem to assume that an interpretation 
is always fixed first, and then applied to all graphs.  In my view the 
formal semantics permits either of these approaches, because it can be 
applied to any interpretation-graph pair.

We probably won't get to the bottom of all this before the RDF working 
group needs to close up shop, so I think we should also assess where we 
stand on the practical issue of finalizing the RDF specifications. 
AFAICT, most of our underlying disagreements have been avoided by 
eliminating language that would cause them to be apparent.  The only two 
places in the specs where I am still aware that these differences become 

1. (ISSUE-148)  The Concepts document currently says:
IRIs have global scope: Two different appearances of an IRI denote the 
same resource.

2. (ISSUE-145)  The Semantics document currently distinguishes between 
"identifies" and "denotes", which I find contrived.  Actually, I've 
already suggested a resolution to this, and I have not heard back from 
you about whether you find it acceptable:

I guess in a separate message I'll discuss the practicalities of 
resolving ISSUE-148, so that we can separate those practicalities from 
the philosophical discussion.

Specific responses on the philosophical issues follow.

On 11/09/2013 10:54 PM, Pat Hayes wrote:
> On Nov 8, 2013, at 11:00 PM, David Booth <> wrote:
>> Hi Pat,
>> Thanks very much for your excellent summary.  Detailed responses
>> below.  (And sorry it takes me so long to write these things up.)
>> On 10/31/2013 02:32 AM, Pat Hayes wrote:
>>> Hi David
>>> Rather than respond point-by-point, I will again try to
>>> summarize. However, there are a few responses that are needed
>>> first:
>>>> ... at least in principle, anything that can be described in,
>>>> say, English prose could instead be described in RDF.
>>> Most emphatically, no. Even if you substitute the most
>>> expressive formal logic available (say, full higher-order modal
>>> tense logic) , this would not be even remotely correct. RDF is so
>>> inexpressive that it cannot manage something as simple as
>>> "Fathers are not mothers." OWL cannot define the idea of an
>>> uncle, and full first-order logic cannot define the idea of a
>>> natural number.
>> I was assuming suitably expressive standard semantic extensions.
> But that begs the question. How do these miraculous extensions
> acquire *their* full meaning? Eventually you will ground out in NL
> prose (perhaps plus some mathematics, etc.) in documents read by
> human readers.

Yes, absolutely.  I assume that there will always be a few IRIs whose 
referent is defined by English prose.  But it seems to me that if the 
Semantic Web is to succeed in its vision, then *most* IRIs must be 
defined by RDF description, in order to be machine processable, for 

>> But regardless, are you claiming that this would fundamentally
>> change the argument?
> Im still not sure what the argument is, but I think this is a basic
> point about how meanings are communicated on the Web. (Or indeed
> anywhere else.)
>> The context and purpose of this assumption is to bar any "then a
>> miracle occurs" steps from the process of IRI definition. Again, a
>> primary assumption is that for scalability, in the vast majority of
>> cases IRI definitions must be provided using description -- not
>> ostension.
> Well, if http-range-14 is accepted, then the vast majority - down to
> several places of decimals after 99% - are going to be determined by
> what we might call http-ostention.

Sure, but that's in regular Web use.  The vast majority of those are not 
so relevant to the Semantic Web.  Do you agree with the assumption that, 
aside from Web page IRIs, for scalability, in the vast majority of 
cases, IRI definitions must be provided using description rather than 

>> In principle, descriptions could either be formal (in RDF) or
>> informal (in English prose, say).  Are you claiming that English
>> prose definitions can somehow avoid underdeterminism that leads to
>> divergence) where RDF descriptions cannot?
> Yes, that is exactly what is meant by saying that English is more
> expressive than RDF.
>> For example, are you claiming that there exists a resource property
>> whose value can be specified in English prose but whose value
>> cannot be specified in RDF (assuming suitably expressive generic
>> semantic extensions)?
> Yes
>> If so, how?
> ?? By writing the English prose. (? I must be missing your point.)
> For example, take how xsd:decimal is defined in
> : "decimal represents a
> subset of the real numbers, which can be represented by decimal
> numerals."

I was talking about in the common case -- not the rare case of defining 
a foundational semantic extension or a purely mathematical notion.  You 
long ago pointed out that only a very few things (such as purely 
mathematical concepts) can be fully defined by description.  So I'm not 
talking about those rare cases, I'm talking about the common case.

I certainly agree that English prose can define something much more 
conveniently than in RDF, and this may postpone the inevitable 
divergence a bit longer than if only RDF were used, but I do not yet see 
a fundamental difference in the trend *toward* divergence.  Some years 
ago you gave a good example of the notion of Pat Hayes, the person.  But 
what *kind* of person notion is it?  The legal entity?  The physical 
body?  The legend?  It sounds like you are now claiming that this 
underdeterminism can somehow be avoided by English prose where it cannot 
be avoided by RDF.   Because if it cannot be avoided then it seems to me 
that we're right back with descriptions causing underdeterminism, 
whether they are RDF or English prose.

>>>> But AFAICT, the trend is inevitably *toward* mismatch as more
>>>> statements are published, assuming that: (a) parties publish
>>>> data independently (without knowledge of each other)
>>> But why would you assume this? Usually, if B is publishing data
>>> using A's IRIs (the only case that is of interest here) then B
>>> will have access to *some* published information which will help
>>> determine what A's intentions were regarding A's intended
>>> meaning. For example, if you use DBpedia IRIs then there are
>>> large pages of information available, in multiple languages. The
>>> entire Semantic Web/linked-data enterprise is predicated on the
>>> idea that IRIs both denote entities and also provide links to
>>> sources of more information about those entities (or, if you
>>> like, more information about what the IRI is intended to denote.)
>>> So the idea of two RDF authors using the same IRI but without any
>>> knowledge of what the owner of the IRI intended it to refer to,
>>> is SW/LD-pathological.
>> It sounds like you misunderstood what I meant.  I was not talking
>> about RDF authors not having access to IRI definitions.  I am
>> assuming that A's IRIs have published definitions that are provided
>> in RDF.  I am talking about the situation where A publishes IRI
>> definitions, and B1, B2 etc. are other parties who independently
>> use A's IRIs in publishing other data.  The assumption is that B1,
>> B2 etc. are all aware of A's IRI definitions, but are not aware of
>> each other (or each other's data).
>> When this occurs, AFAICT the trend will inevitably be toward
>> mismatch as B1, B2, etc. publish more data, even if each of those
>> datasets is *individually* fully compatible with A's IRI
>> definitions.
> I am much less convinced of this than you are. Of course this does
> happen in practice, but there is also a lot of awareness of the
> dangers and people trying to take care to avoid it. I acknowledge the
> possibility but push back on the "inevitability".

But how can it be avoided without requiring the IRI owner to constantly 
monitor all RDF that is asserted involving that IRI, check for 
inconsistencies, and periodically update the IRI definition?    (And 
that does not seem at all practical to me.)  Presumably RDF authors who 
use the IRI will make statements that are *not* entailed by the IRI 
definition, as otherwise it would be rather pointless to make the 

>>>> If the problem is disagreement then yes, you would have to
>>>> choose between the source graphs.  But if the problem is
>>>> divergence then you have to do some more work -- resource
>>>> identity splitting -- but can still use both source graphs
>>>> after splitting.
>>> Changing the IRIs in a graph gives you a different graph. So you
>>> would not be using both source graphs, but some modification of
>>> the source graphs. And you would be obliged to *not* use - that
>>> is, to reject - at least one of the source graphs, when they are
>>> mutually inconsistent.
>> Are you just dotting i's and crossing t's here?  The point is that
>> the original graphs are *not* rejected, but you're right that they
>> are not merged **as is** either.
> If they are "changed", then that is a form of rejection. Let us be
> precise (the WG spent months getting this right): graphs cannot
> change. What you can do is reject one graph and use a modified one,
> but that modification is a *different* graph.

Of course it's a different graph!  That's the whole point of the 
transformation that derives it!   But if you're insisting on calling 
this "a form of rejection" then it sounds like you're still missing the 
main point here.  Did you understand the rest of the paragraph, below, 
where I explained the difference in more detail?

>> The graphs are used by systematically transforming one or both of
>> them (by properly substituting IRIs -- a process very similar to
>> skolemization) and then merging the result, so as to join and use
>> the relevant information content of *both* graphs. This is
>> fundamentally different from disagreement, in which relevant
>> information content must be *discarded*. By "relevant information
>> content" I'm talking about the information content of the graph
>> other than the particular choice of IRIs that were assigned -- a
>> kind of graph isomorphism (though more general than the notion of
>> isomorphism that is defined in the RDF Semantics, which is only
>> isomorphism over bnode relabeling rather than IRI relabeling).

Do you see the practical difference between divergence and disagreement? 
  In one case the relevant information content is retained; in the other 
it is discarded, because it is assumed to be wrong.

>>>> ... RDF data does not generally describe the real world, it
>>>> describes a particular *conceptualization* of the real world
>>> It describes the world *using* a conceptualization (is there any
>>> other way to describe anything?). It does not (usually) describe
>>> the conceptualization.
>> Maybe we're just quibbling over words here, but I don't see how RDF
>> data that models the world as flat could possibly be describing the
>> real world, since the real world is *not* flat.
> It is locally, approximately flat. If your interpretation of "flat"
> allows for this locality and approximation, then it can be true to
> describe the world as flat in that sense.
>> In essence, unless RDF is 100% accurate then it isn't describing
>> the real world, is it?
> Of course descriptions may be loose or approximate or incomplete, and
> yet still be true. I have worked as part of a framing crew, where
> someone on a ladder will call down to someone else operating a chop
> saw, and say: "94 and three quarters, fat!" and then complain when
> the cut 2x4 he is passed does not fit the frame. These utterances are
> about the real world and have truth values, even if they are not
> accurate to thousands of an inch or fail to mention the species of
> the tree from which the timber was cut.
>> It could just as well be claimed to be describing my left earlobe.
>>>> false graphs aren't very useful, because they entail
>>>> everything
>>> Just as a technical point: *logically false* graphs 
>>> contradictions, false in *every* interpretation  entail
>>> everything. Mere falsity does not get you quodlibet.
>> Interesting point.  Do you think that has any impact on this
>> analysis? If so, how?  The context of that statement was just
>> pointing out that false graphs aren't very useful.
> Well, it means that the case for inutility has to be re-stated.

Fine, then please re-state it some other way that you prefer if you 
want.  My point was that false graphs are not very useful.  I may not 
have chosen a good way to explain *why* they are not very useful, but as 
long as we agree that that is the case then we don't have to get bogged 
down in the details of explaining why.

>>> ---------
>>> There are two substantive points of disagreement between us, and
>>> one complete mismatch (divergence?) where I fail to understand
>>> what you are saying. Let me deal with the two points first.
>>> 1.  The reality thesis: that the real world is one of the
>>> satisfying interpretations, and data is (usually) about the real
>>> world. I find this obvious, so obvious indeed that it should not
>>> even need to be said. You apparently find it either mistaken or
>>> meaningless, and in any case think it is misleading as a guide to
>>> intuition. I am not sure how to persuade you to my way of
>>> thinking, but let me ask you: if all this linked data is not
>>> about reality, what do you think it *is* about?
>> Informally (and imprecisely) we can say that published RDF data is
>> about the real world.
> It might be informal, but it is true, and fundamental.
>> But in general I think it would be more precise to say that it is
>> *directly* about a particular conceptualization of the world
> No, It is not ABOUT a conceptualization. (Well, not usually.
> Arguably, the SKOS defining documents are about a conceptualization,
> for example, but then they *use* a different conceptualization.) This
> is a very basic point in discussing semiotics at all: we have to keep
> a clear distinction between what a description is *about*  - what it
> describes - and *how* it describes it, the conceptualization it
> uses.
>> , and only *indirectly* about the real world, through an informal
>> and unspecified correspondence between that conceptualization and
>> the real world.
> To use a conceptualization to describe X is not describing something
> other than X. It is describing X *as being* the thing so described.
> For example, I can think of you as a dynamic process involving a host
> of chemical events inside cells, rather than thinking of you as a
> person. But it would still be *you* I am thinking about, whichever
> way of thinking *of you* I choose to adopt.
>> For example, some RDF data may describe a flat-earth
>> conceptualization of the world, and that flat-earth
>> conceptualization corresponds in some way to the real world.  RDF
>> data does not *directly* describe the real world, because the real
>> world is *not* flat.
> It describes it through an imprecise or approximate
> conceptualization. But approximation does not change the object of
> the description.

It sounds like we at least agree that such "description" is indirect, 
via a conceptualization, and we're merely quibbling about whether the 
word "describe" is appropriate.  But maybe there's something deeper to 
it.   In terms of RDF semantics, what exactly do you mean by "the object 
of the description"?   Do you mean the referent of an IRI in an 
interpretation?  Because if you don't, then I *really* don't know what 
you mean.

>> Specifically, such RDF would ascribe properties to the earth that
>> the earth does not possess.
>>> And why do we find it useful, if it does not provide us with
>>> information about the actual world?
>> It is useful because, when we feed that data into our applications,
>> our applications produce the desired outputs.  It doesn't matter
>> what the data is about as long as it produces the desired outputs.
>> I'm not saying that to be flippant, but to emphasize that the whole
>> notion of the real world is completely irrelevant to machine
>> processing.  Machines just push the bits around, and if they come
>> out with the answers that we want, we're happy.  What really matter
>> is usefulness of the data.  But I agree that, all other things
>> being equal (which they aren't), conceptualizations are *much* more
>> useful when they more closely resemble the real world.
> We kind of agree and kind of don't agree here, but it would take too
> long to sort out the threads. (Basically, I think that utility almost
> always only arises from truth, in this kind of a context. With
> exceptions such as deliberate lying or confabulation to achieve a
> political or military end, but this goes outside 'normal' RDF data
> usage.) Just in passing, conceptualizations do not *resemble* the
> world. They are ways of thinking about the world. Arguably, we cannot
> think about or describe the world at all except through some
> conceptualization or other. Even quantum physics has its
> conceptualizations.

Right, but my point was that the conceptualization is *not* the same as 
the real world.  The map is not the territory.  RDF generally 
"describes" (though you may prefer a different word here) the map -- not 
the territory.  We may then interpret that description as being about 
the real world, or we may interpret it as being about some fictional 
world that is relevant to a particular application.

>>> Are the records of the transactions in your bank account about
>>> your actual wealth? Would that change if the bank started using
>>> RDF?
>>> Your objections to the idea include the observation cited above
>>> about conceptualizations. Yes, of course data is stated *using*
>>> a conceptualization, just like all assertions in every language
>>> or formalism. But that does not make it any less about reality.
>>> It really is a fact about the real world that Hilary and Tensing
>>> climbed Everest in 1953; that we conceptualize the world here in
>>> terms of people and mountains does not make this any less true. I
>>> am not sure what the point of your "toucan" example is, but
>>> apparently the real world can satisfy both the bird assertions
>>> and the website assertions, by appropriate choice of an
>>> interpretation mapping. (If the complete set of assertions is
>>> inconsistent then of course nothing can satisfy it.) Your third
>>> point concerned approximations and idealizations, such as the
>>> flat-earth geography of road maps. But examples like this do not
>>> argue against the reality thesis. An approximate or idealized
>>> description of X is still a description of X. Bear in mind that
>>> if some RDF can be satisfied by an approximation or
>>> simplification of the real world, then it can also be satisfied
>>> by the more complicated real world, since one can add (an
>>> infinite amount of) structure to an interpretation freely without
>>> making any RDF triples false. (This is a consequence of RDF being
>>> a positive logic without negation.) The map example is quite
>>> instructive, as quite a lot of geolocation information (eg
>>> lat/long coordinates) is in fact describing spherical space
>>> rather than flat space, even though we project it onto flat
>>> surfaces.
>> First of all, let me distinguish between two cases, to be sure that
>> we are talking about the same case.  One case is where the
>> conceptualization is carefully crafted to acknowledge the fact that
>> it is an approximation, so it might say, for example, something to
>> the effect that "pi is greater than 3.1 and less than 3.2".  I'm
>> not talking about this case.
>> The other case is where the conceptualization in effect says "pi
>> equals 3.1".  In other words, it asserts something that may be good
>> enough for many applications, but in fact it turns out to be false
>> when scrutinized in detail in the real world.
> Well, is it false? That depends on what exactly "pi" and  "equals"
> are interpreted to mean. If you mean the actual mathematical terms
> then yes, it is false.

Yes, I meant the mathematical terms.

> But if all quantities are understood here to
> be rounded to one place of decimals then this is true.

That's the *other* case, not this one.  I was specifically 
distinguishing between these two cases: In case #1, the RDF has been 
carefully crafted to be absolutely true in the real world.  In case #2, 
it is *not* true in the real world (because pi is *not* exactly 3.1 .)

>> This is the case that I'm talking about, and I used a flat earth
>> conceptualization as an example, because it is useful for some
>> applications but clearly wrong in terms of (directly) describing
>> the real world.
> I disagree about its being clearly wrong, and I disagree about most
> of the data being based on it. In fact, geolocation data uses one or
> another virtual spheres projected from BPS satellites, and yet we use
> it together with 'flat earth' data such as road mileages between
> cities without this concpetualiztion mismatch being an issue.

I have stipulated that in case #2, we are talking about the mathematical 
notion of pi, equality and 3.1, so it *is* clearly wrong.
Are you therefore claiming that case #2 does not exist?  If not, and you 
are just quibbling with my choice of example, then please pick a 
different example for case #2 that you like better, so that we can move 
forward on the substance of this discussion.

>> Furthermore, AFAICT this second case is where divergence inevitably
>> leads (barring splitting, which mints new URIs).  The reason is
>> that divergence leads to a situation where one graph A in essence
>> asserts "X P V" and another graph B asserts "X P NotV"
> ? Do you mean X NotP V ?

No, I meant a value of P such that "X P NotV" is inconsistent with "X P 
V" (assuming that P cannot be both V and NotV).  I realize this is an 
odd way of writing it, but I wrote it that way to make it easy to see 
that X's definition may not have originally constrained the value of P, 
and later one party may assume a value V and another may assume a value 
NotV, thus causing divergence.

>> , which means that from some perspective, one of those graphs is
>> *wrong*.  And yet, within each of those graph's intended
>> interpretations, the graph is *not* wrong, it is true.  I.e.,
>> within each graph's *conceptualization* of the world, the graph's
>> statements are consistent with that conceptualization.
> What is a *graph's* conceptualization? Graphs use IRIs to refer, and
> IRI reference transcends graphs. It makes sense to speak of a
> particular IRI vocabulary as comprising a conceptualization, but not
> a graph as having one in isolation.
> Thinking about this later, maybe what you mean here is the
> interpretations of the graph intended *by the author of the graph*,
> ie something like what the author *had in mind* when they composed
> and published the RDF, what is was they were *trying to say*.

Yes, exactly.

> In
> which case, yes indeed two graphs may have divergent interpretations
> in this sense. But that is not what RDF Semantics is trying to
> capture: it is not about what an RDF author is *trying* to say, but
> what *actually gets said* by the RDF they publish.

What *actually* gets said under the formal semantics is just an RDF 
graph that is either: (a) false under *every* interpretation; or (b) 
true under some interpretations and false under others.  Nothing more.

AFAICT any attempt to impart meaning *beyond* that will depend on the 
assumptions and biases that we bring in -- i.e., the interpretations 
under which we choose to evaluate it.   And IMO when someone publishes 
an RDF graph, the most sensible way to understand that act (under normal 
circumstances) is to view the publisher as asserting that the graph is 
*true*.  And that raises the obvious question: "Under what 
interpretations?", which takes us back to the question of the author's 

IMO, what an RDF author is *trying* to say is certainly relevant.  The 
point of the RDF Semantics is to help RDF authors communicate with RDF 
consumers.  The best communication is achieved when the consumer best 
understands what the author intended.  So although you can certainly 
claim that (in a narrow sense) the RDF Semantics was never intended to 
be used this way, the point is that it *is* relevant *and* useful to 
think of the RDF Semantics this way.

> This might be a
> good deal less than what they were trying to say (and so have a lot
> more interpretations than are in the set of "intended
> interpretations", which is something that nobody but the author, and
> maybe not even her, can possibly have any access to.)

But hold on, you're throwing the baby out with the bath.  Even if you 
cannot get inside the author's head, as a starting point you *can* 
consider all of the graph's *satisfying* interpretations.  It is 
reasonable to assume that the author's *intended* interpretations are at 
least a subset of the satisfying interpretations.  Furthermore, that set 
can be further reduced by considering only the subset that is consistent 
with the interpretations of well known IRIs whose interpretations are 
already fixed (by some "miracle" step).  Finally, even if you cannot 
*formally* nail down the author's intended interpretations, it is still 
a useful guiding intuition (to some, at least, since we know that 
people's intuitions vary), because it corresponds better, to the idea of 
understanding the author's intent, than the idea of a single, shared 
"real world" interpretation.

>>> To say that some assertion is about the real world, or that it
>>> is factual, is not to claim that it is in some metaphysical sense
>>> the final truth or the definitive description, or that it is the
>>> last word, or that its truth has to have ended science. It is
>>> just saying that it is true.
>> I think what you mean here is that there may be *additional* facts
>> that are also true.  Right?  If you are telling me that there are
>> various different notions of truth, then you'll have to explain
>> more.
> No, I want a single definition of truth. But I don't want to have
> that notion lumbered with extra baggage of being completely accurate
> or completely complete in some impossible way.

I have no idea what you mean when you talk about lumbering truth with 
"extra baggage", or truth being "completely accurate" or "completely 
complete".  I assumed we were talking either about the objective, 
pre-formal notion of truth -- a binary distinction -- or about the RDF 
semantics assignment of a truth value to a graph-interpretation pair, 
which is also a binary distinction.   In either case it just true or 
false, isn't it?  Can you explain what you mean by "extra baggage" or 
"completely accurate" in this context?

>>> You say:
>>>> ... The "real world" interpretation is largely irrelevant --
>>>> both to the formal semantics and to understanding how the
>>>> Semantic Web *actually* works.
>>> I strongly disagree. Many IRIs have fixed interpretations in the
>>> actual world,
>> I was wondering when you would bring that up.  :)   I partially
>> agree, but I think the degree is far different.  I would say that a
>> *few* IRIs have fixed interpretations -- meanings that are fixed by
>> ostension.  But the vast majority are defined by description, and
>> those by necessity are far less fixed.
>>> determined by all kinds of social, technical and linguistic
>>> conventions and meanings entirely outside RDF.
>> We need to be careful about the assumptions that we make when
>> stepping outside of RDF, to avoid "then a miracle occurs" steps
>> whenever possible. My assumptions are:
>> - A very few IRIs are defined by ostension.  (This is one "then a
>> miracle occurs" assumption.)
>> - All other IRIs are defined by description.  For simplicity, let's
>> assume either English prose or RDF.
>> - For the most part, English prose definitions do not fix the
>> interpretations significantly more than RDF definitions, because:
>> (1) machines cannot understand English prose, and the Semantic Web
>> is intended for machine processing; and (2) an English prose
>> definition could in principle be written instead as RDF (assuming
>> suitable standard semantic extensions).
> OK, I definitely do not accept this last assumption. Your point (2)
> is just flat wrong, unless the extensions are themselves defined
> using natural language (which makes the point vacuous),

Yes, I assume that the extensions themselves are defined using some 
"miracle" step, for boot-strapping -- either via ostension or natural 
language.  This does *not* render the point vacuous.  The assumption is 
that the vast majority of IRIs thereafter are defined mainly by RDF, 
because (a) sufficient foundational IRIs have been defined using natural 
language or ostension, to enable rich-*enough* RDF definitions; and (b) 
RDF enables machine processing.   Maybe I'm wrong to make this 
assumption, but I don't see how the Semantic Web could scale in a 
machine processable way without it.

> by Goedel's
> theorem. No computable axiom system can even define the integers.
>> I know you disagree with point #2 of that last assumption, and you
>> may well be correct about that in a theoretical sense, but the
>> question is: is there really a big difference in our ability to
>> avoid unintended interpretations via English prose definitions
>> versus RDF definitions?
> Yes.
>> And if so, is that difference big enough to override the effect of
>> point #1?
> Yes, I would say. (Although of course English *can* also fail to
> describe or define meanings adequately.) English prose (all natural
> languages) relies on a huge mass of terms and ideas which are
> connected to the real world through our senses, perhaps by ostention
> but more likely through reading other NL prose. No axiomatic formal
> system can reproduce this kind of connection, but it can use it and
> extend it.

Yes, and that's exactly what I assume happens for a *few* IRIs.  But it 
sounds like you are assuming that we will *mostly* rely on English prose 
definitions, and AFAICT that won't work very well for the Semantic Web, 
because Semantic Web applications don't understand English prose, and 
the whole point of the Semantic Web is to automate.

>> If we assume that English prose descriptions fundamentally fix the
>> interpretations substantially more than RDF descriptions, then it
>> represents a second "then a miracle occurs" step in the process --
>> a step that surely is not scalable for the Semantic Web in the same
>> way that RDF descriptions are scalable.
> The point is not that the semantic web machinery can take part in the
> miracle of human communication, but that it can as it were ride upon
> it. The machinery will only 'know' that part of the meaning of IRIs
> that can be reflected in the formal model theory, but the conclusions
> it draws will be expressed using IRIs which have a richer meaning
> than the machinery itself can make use of, but which will be
> available to human readers of the mechanically drawn conclusions.

Yes, exactly.  But if IRI definitions are expressed in English prose 
instead of RDF, then the machinery will not be very successful in 
drawing useful conclusions.

The degree to which the machinery can draw useful conclusions is 
*directly* related to the degree to which IRI definitions (and other 
relevant information) are expressed in RDF, rather than English prose.

> This works when the mechanical inference does not destroy or modify
> any of that richer meaning, of course; but if the human notions of
> truth can be expressed in model-theoretic terms (even if they use
> constraints that would be miraculous if we were limited to RDF) then
> this will always be the case. Exceptions to this happy picture might
> for example include social constraints on the appropriateness or
> otherwise of certain descriptive vocabularies. The Turkish government
> gets very upset when the anyone refers to "genocide" of Armenians,
> even if this is factually correct. RDF semantics does not deal with
> such social and diplomatic delicacies in language use, however.

Agreed.  I think we can ignore that case.

>>> We still want to be able to use RDF to describe these referents.
>>> For example, I am a consultant on a project
>>> ( to add RDF markup to images.
>>> These RDF descriptions use IRIs which identify (and in the RDF
>>> refer to) images, regions in the images, people and places and
>>> colors and objects described in DBpedia and many other real (no
>>> scare quotes) things in the real world. None of these denotation
>>> mappings are specified by RDF descriptions, and most of them
>>> could not be. Most  I would claim, virtually all  RDF linked
>>> data uses IRIs like this to refer to real things. It is
>>> centrally important that the formal semantics works with such
>>> identifying IRIs.
>>> 'Edmund Hilary climbed Everest in 1953' says something true about
>>> the actual, real, world. It expresses a fact. Just a mundane,
>>> simple bit of data. So, how is this factuality of this fact
>>> related to model-theory semantics? By the actual, real, world
>>> being one of the satisfying interpretations of it. Because if the
>>> real world was not a satisfying interpretation of this sentence,
>>> then it *couldn't possibly* be true (in the real world.)
>>> But we can, if you like, simply agree to disagree about this, as
>>> it has no direct bearing on the basic point we have been arguing
>>> about, which is...
>>> 2.  The idea of an IRI denoting something "in a graph".  Your
>>> gloss on this phase, as I now understand it from your email (the
>>> first time you have explained your intended meaning) is as
>>> follows: you take all the interpretations which satisfy the graph
>>> (and there will be different such sets for different graphs, of
>>> course) and then you ask, what does the IRI denote in those
>>> interpretations? And that is what the IRI denotes "in the graph".
>>> (Do I have that right?)
>> Yes, exactly right.
>>> But that does not define anything, because for any consistent
>>> graph G, and any IRI U in that graph, there are interpretations
>>> which satisfy G and in which U denotes things different from what
>>> it denotes in other interpretations satisfying G. There is no
>>> graph which 'pins  down' the interpretations of the URIs which
>>> occur in it in the way that your definition requires.  (Here is a
>>> simple proof. Let x be something which is not an IRI. The
>>> interpretation I with universe {x} and IEXT(x)={<x,x>} and I(u)=x
>>> for every URI u, satisfies G. The Herbrand interpretation H of G
>>> also satisfies G. But H(U) = U =/= x = I(U), by construction.
>>> QED.) In fact, one can make a stronger statement: truth in an
>>> interpretation does not depend on the identity of the referents
>>> of IRIs *at all*, because one can take *any* satisfying
>>> interpretation and produce another isomorphic one with the
>>> identities permuted in any way one likes, as long as the IEXT
>>> mappings are permuted to match. (In fact, this applies to *any*
>>> axiomatizable, complete formal logic, no matter how expressive.)
>>> In a nutshell: model theory does not determine reference.
>>> This should not be too surprising, actually, if you think about
>>> how model theory is defined. The very definition of
>>> interpretation presumes complete referential freedom: any IRI can
>>> denote anything. And truth is determined solely by how those
>>> things stand in relations to one another. The entire apparatus of
>>> model theory makes no reference to the *actual identity* of the
>>> things in the universe being described. So creating real
>>> constraints on reference - attaching, as it were, a name to a
>>> thing - has to be done by other means. In practice, we rely on
>>> notions of naming and reference already in use in the larger
>>> world (as I did when using "Everest" to refer to the highest
>>> mountain, and how ImageSnippets does when using
>>> '' to refer to the class of human beings)
>>> and sometimes on predefined mappings (as we do when fixing the
>>> referents of literals using datatypes) and perhaps even by
>>> ostention (arguably, http-range-14 can be seen as declaring HTTP
>>> GET/200 to be a form of ostention.) And this all works quite
>>> nicely (a lot of the time) because we can all (more or less)
>>> agree on what these referring names actually refer to, at least
>>> well enough to transfer meanings successfully by using them as
>>> referring names in sentences.
>>> So, as I believe I have said several times, phrases such as
>>> "interpretation of an IRI in a graph" are not meaningful. It is
>>> not that this is a different perspective on model theory, or an
>>> alternative viewpoint. It is that it, quite literally, does not
>>> mean anything.
>> Yes and no.  I fully agree with your proof, and the fact a Hebrand
>> interpretation will always satisfy the graph, so let's not get hung
>> up on that.
> Well I agree lets not get hung up on it, but let us also not simply
> ignore it, as it refutes your claim.

No, it doesn't, as the next paragraph already explained.

>> But you have left out two key things.  One is the existence of
>> *some* IRIs that -- as you pointed out above -- *do* have their
>> interpretations fixed, presumably by ostension.  Those will (by the
>> rules of entailment) cause the possible interpretations of *other*
>> IRIs to be reduced also -- usually still not uniquely, but
>> nonetheless reduced.  A simple example is if <http://example/aa>
>> has a fixed interpretation, and <http://example/aa> owl:sameAs
>> <http://example/AA>, then by entailment <http://example/AA> also
>> has a fixed interpretation.
> Yes, of course. (In an old publication, I used the analogy of tying
> down a fabric to a surface by attaching it at enough points and
> relying on the strength of the fabric to hold the rest of it.) This
> is exactly how meanings can be transferred across entailments. But
> this relies upon the interpretations of occurrences of IRIs being
> *independent* of the particular circumstances of their occurrence. If
> (two instances of) the same IRI in two places can mean different
> things, then these fixed interpretations cannot be preserved through
> inferences.

But again, don't throw the baby out with the bath.  I think it is 
reasonable to assume that there are a *few* IRIs whose interpretations 
are globally fixed, by a "miracle" process.  But that does *not* mean 
that we need to make that assumption of *all* IRIs.  And AFAICT, we 
should assume that most IRIs are defined by RDF, so that the definitions 
are machine processable.

>> The second is the fact that every application that consumes RDF has
>> a preconceived set of *assumed* interpretations.
> I simply do not agree with this. In fact I do not think it is
> coherent. How would this preconceived set yield any difference in
> behavior, for a conforming RDF engine?

I don't know what you mean by "difference in behavior".  Different from 
what?  The set of assumed interpretations further restricts the set of 
satisfying interpretations for a graph, to those interpretations that 
are relevant to the application.

Think of an RDF application as a machine that takes RDF input and 
performs some action as a result of its input.  The action depends on 
the application.  For example, if it is an access control app, then it 
might interpret <> as denoting you -- not your 
mailbox -- and it might grant you access when you swipe your card at the 
door, and the card reader sends a blob of RDF to the access control app. 
  In that example, the application's assumed interpretations were about 
people and doors, and who is allowed to open which doors.  The assumed 
interpretations did *not* include interpretations that map 
<> either to itself (as a Herbrand interpretation) 
or to a mailbox, because the notion of a mailbox does not exist in that 
app's conceptualization of the world.

Apps will differ in the degree to which they further restrict a graph's 
satisfying interpretations.  Some, such as a simple reasoner, won't 
further restrict them at all.  Others, such as the door control app, 
will severely restrict them.

>> That set of assumed interpretations represents the particular
>> conceptualization of the world in which that application operates.
> No. Conceptualizations are not graphs and they are not sets of
> interpretations.

A conceptualization, as I've been using the term, corresponds to a set 
of interpretations in the formal semantics.

>> For example, in the toucan example, one application uses a
>> conceptualization of the world involving web pages, and in that
>> conceptualization, <http://example/toucan> becomes fixed to a web
>> page -- not the string "http://example/toucan" (as in a Hebrand
>> interpretation), and not the bird either, but the web page.
>> Similarly, in a different application whose conceptualization of
>> the world involves birds (but not web pages), the interpretation of
>> <http://example/toucan> becomes fixed to a bird.
> As far as I understand the point of this example (and I am not sure
> that I do), both of these hypothetical applications are simply
> drawing invalid conclusions about what the IRI means. (The web page
> app is assuming that something that might be a bird is in fact a web
> page, and oppositely for the other case.) Which is fine: they are
> free to do so. But the semantics has nothing constructive to say
> about that, other then that they are indeed drawing invalid
> conclusions.

Why do you say the conclusions are invalid?  In one case, the bird app 
uses one set of RDF (about birds) to conclude that the IRI denotes a 
bird -- following all of the rules of proper entailment -- and in the 
other case, the web page app uses a different set of RDF (about web 
pages) to conclude that the IRI denotes a web page, again following all 
of the rules of proper entailment.

As a simple example, suppose that somehow the referents of fixed:Bird 
and fixed:WebPage are rigidly fixed as being the classes of birds and 
web pages, respectively, and rdfs:subClassOf rigidly denotes the 
subclass relationship.  Now suppose Ian Davis's RDF says something to 
the effect:

   # Graph G
   <http://example/toucan> a ex:ColorfulBird .
   <http://example/toucan> a ex:HttpResponder .

and suppose the bird app uses this additional RDF:

   # Graph Gb
   ex:ColorfulBird rdfs:subClassOf fixed:Bird .

but the web page app uses this additional RDF:

   # Graph Gw
   ex:HttpResponder rdfs:subClassOf fixed:WebPage .

Then, through entailment, the bird app concludes that 
<http://example/toucan> denotes a bird, while the web page app concludes 
that it denotes a web page.  Isn't such entailment valid?

>> Finally, if we back up one step from the idea that each consuming
>> application has a set of assumed interpretations, we are back to
>> the idea that each graph has a set of intended interpretations,
>> which represent the set of assumed interpretations that the graph
>> author intended to support.The net effect is that it *is*
>> meaningful to talk about the idea of an interpretation of an IRI in
>> a graph.
> No, we are not back at that idea. In fact, I fail to understand how
> you even see any connection between these two observations and this
> central claim of yours. Suppose for the moment that some IRI meanings
> are determined by ostention and that some applications draw logically
> invalid conclusions from RDF.

But in the example above, the conclusions are *valid* according to 
standard entailment rules, aren't they?  Or have I made a mistake somewhere?

> How does all of this imply that it is
> meaningful to speak of an interpretation of an IRI in a graph?

Within the merge of G+Gb, <http://example/toucan> denotes a bird, 
whereas within the merge of G+Gw it denotes a web page.  That is a 
normal and -- to some anyway-- very natural way of thinking and talking 
about that IRI's meaning.

> By the
> way, the point made in my earlier reponse, that any referent can be
> simply replaced by another without changing truth values as long as
> the extension mappings are redefined to match, applies to *any*
> referent not itself fixed by ostention or in some other rigid way.

Yes, I know that.

>> In fact, I would conjecture that that is how many people in the
>> Semantic Web think of IRI meaning, whether they are conscious of it
>> or not.   It's really a very sensible and intuitive way to think of
>> it, IMO
> I believe that I have shown quite rigorously that it is incoherent
> and meaningless.

No, but you have shown that you have not yet grokked what I'm trying to 
say.  :(  And I admit that if your natural intuition about this stuff is 
to think in terms of a single "real world" interpretation, then what I 
am trying to say may seem bizarre.  But AFAICT it is entirely coherent, 
and in fact *useful*, as it provides a way of thinking about the formal 
semantics that more closely aligns with actual practice.

> But even if I am wrong about that, it is certainly
> in violation of the RDF Semantics specifications as published, which
> define an interpretation to be a mapping from IRIs to referents, not
> from IRI occurrences in graphs to referents.

Clearly you're still misunderstanding what I'm saying, because I have 
*never* said that an interpretation is a mapping from IRI occurrences in 
graphs to referents.  We have already gone over this several times and 
confirmed that the formal semantics I am assuming are *exactly* RDF's 
model theoretic semantics.  The difference is *only* in how I am using 
and thinking about the semantics.   The formal semantics are the *same*.

Look, there is a big difference between saying "an interpretation is a 
mapping from IRI occurrences in graphs to referents" and saying 
"different interpretations may be *applied* to different graphs".  They 
are *not* the same thing.  And the reason for applying different 
interpretations to different graphs is because: (a) different graphs 
have different intended interpretations; and (b) different consuming 
applications have different assumed interpretations.

> So those who do think
> this way are misunderstanding the RDF specs and may be in danger of
> writing non-conformant apps.
>> , though as we know, people's intuitions differ.
>>> ---------
>>> Now the place where I fail to understand what it is you are
>>> saying.
>>> At the end of your email you list all the advantages of an
>>> "other way" of approaching model theory. But as far as I can
>>> tell, this "alternative" is simply standard model theory.
>> I agree!  That's what I've been saying all along.  But AFAICT,
>> based on your other comments, it does seem to involve a slightly
>> different way of *thinking* about the standard model theory.
>>> For example:
>>>> The other way to think of the RDF Semantics is in terms of
>>>> *multiple* interpretations
>>> This is the only correct way. As I have said to you before, *of
>>> course* we think in terms of multiple interpretations. That is
>>> the entire point of defining the notion of interpretation. The
>>> very definition of entailment refers to multiple
>>> interpretations.
>> Well yes, but see your next statement.
>>>> , instead of attempting to assume or impose a single "real
>>>> world" interpretation.
>>> Well, it is fine to assume that the real world is *one*
>>> interpretation, but nobody has ever suggested "imposing" a
>>> single interpretation. Certainly, nothing in the RDF Semantics
>>> document speaks of anything like this.
>> Uh . . . but that is *exactly* what this offending "intuitive
>> introduction" did (before it was removed) and why I insisted that
>> it needed to be scoped to an interpretation: [[ An RDF graph is
>> true exactly when: 1.  . . . etc.
> No, this was saying that the graph is TRUE when, etc.. That does not
> appeal to the model-theoretic notion, but to a pre-theoretic notion
> of what it is for a sentence to be true. It was intended to convey
> the intuition that the model theoretic acount of truth was, when
> stripped of its mathematical jargon, a very close fit to this
> pre-theoretic notion. Which was the point of the (now deleted)
> section in the semantics.

But it *must* appeal to the model-theoretic notion of the sentence being 
true, as it would not make sense to appeal to any other notion.  A 
sentence in a formal language *only* has meaning in the context of its 
defined semantics.  And in the RDF case, the notion of a graph being 
true **only exists** relative to one or more interpretations.

We *can* say: "Graph G is true.  Therefore, what interpretations must we 
be assuming?"  Answer: G's satisfying interpretations.  We *cannot* 
meaningfully claim that a graph is true in the absence of any notion of 
an interpretation under which that graph is true.

It seems that a key difference in our assumptions is the intuition of 
what it means for an RDF author to assert that a graph is true.  My 
intuition is to understand this as meaning that the author is asserting 
that there exists an **intended interpretation** under which G is true. 
  Whereas it sounds like your intuition is to understand this as meaning 
that the graph is true under a particular interpretation that is known 
to all parties and corresponds to the real world.

>> ]]
>> And you argued:
>> [[ To say that a graph (or any other assertion or sentence) is
>> true, is to say that when it is interpreted *in the actual world*,
>> its truth-value is true. ]]
>> That is *exactly* the kind of implicit assumption
> What "implicit assumption" do you think is being made here?

That the graph is interpreted according to a particular interpretation 
that corresponds to the real world.  That's how I'm parsing the phrase 
"when it is interpreted *in the actual world*" in the above sentence. Is 
that what you mean?  If not then I don't know what you mean.

An interpretation is a mapping from IRIs to things and the relationships 
between those things.  Hence, when I say that an interpretation 
*corresponds* to the real world I mean that those relationships are the 
actual relationships that hold between those things in the real world.

> That
> there is only one interpretation of the graph? No, that is not being
> assumed.

Hmm, maybe the misunderstanding here is that you are saying that there 
*exist* multiple interpretations, but the only one that matters is the 
one that corresponds to the real world?

> That there is only one world that we all inhabit? Well, yes,
> I do assume that,


> but I will not be drawn into a debate about that in
> this forum.

Agreed.  That would be pointless.

>> that I think is misleading, and why I have been trying hard to
>> explain how a "multiple interpretations" view of the semantics is a
>> slightly different view from thinking of the semantics in terms of
>> an "in the actual world" interpretation.
> I wish I did not have to keep saying this, but you really do need to
> get a better grasp of the basic foundational ideas of model theory.
> It is intended to be a formal theory of a natural phenomenon in the
> real world, which is that sentences really can express truths. The
> theory is inherently about "multiple interpretations", especially
> when it is used to characterize valid entailment relationships. To
> say that a sentence is true when what it says about the actual world
> is correct, is not to deny the inherent multiplicity of
> interpretations. The first is a statement about nature; the second is
> a statement about the formal apparatus of model theory.

It would be quite surprising if, after all these years, I had managed to 
miss some basic foundational idea of model theory, but I suppose 
anything is possible.   But AFAICT what you are saying is something like 
the following.  If an RDF graph is true in a particular interpretation 
according to the formal semantics *and* if that interpretation happens 
to correspond to the real world -- meaning that for all of the IRIs in 
the graph, the denoted relationships are in fact the relationships that 
exist in the real world -- then you can say that the graph is true not 
only in a formal sense, but also in a pre-theoretic sense.  (I've 
ignored bnodes for the moment, as I didn't want to belabor the 
explanation with a bunch of "there exists . . . " complication that 
would not fundamentally change the point.)  If I've understood you 
correctly, that is the sense in which "sentences really can express 
truths".  Is that correct?

If I've understood that correctly, then that is all fine and dandy, and 
from a theoretical perspective that's pretty cool.  But from a practical 
perspective in the Semantic Web, it only seems *partially* relevant, 
because: (a) different people do not have the same notion of the real 
world; (b) *usefulness* matters more than real world truth; and (c) RDF 
is for machine processing, and machines don't care about truth.

Take the notion of real world truth, instead substitute **intended 
interpretations**, and compare.  I.e., instead of saying:

   "for all of the IRIs in the graph, the denoted relationships
   are in fact the relationships that exist in the **real world**",


   "for all of the IRIs in the graph, the denoted relationships
   are in fact the relationships that exist in the **intended

This sets a lower bar than real world truth.  It seems to me that it is 
more reasonable to expect an RDF author to know what are his/her 
intended interpretations than to know real world truth.

>> The intuition seems to be different.
>>>> By this I mean, for example, that:
>>>> - Two different graph authors may have different sets of
>>>> intended interpretations in mind when they publish their RDF
>>>> graphs, and the same URI may indeed denote different resources
>>>> in those interpretations.
>>> Different sets of interpretations in mind, yes, of course
>>> (standard). URIs denoting different things in different
>>> interpretations, yes of course (standard). URIs denoting
>>> different things in different *sets* of interpretations, yes, if
>>> we are talking about sets of interpetations an author *has in
>>> mind*. But URIs denoting things in a set of all interpretations
>>> which satisfy a given graph? No, for the reasons described above.
>>> That idea is incoherent.
>> I've addressed that above.
> Actually you have not. And later in the message, you claim to be
> talking about set of interpretations the author *has in mind*, not
> sets which satisfy a given graph. This shifts the ground of the
> discussion in a crucial way.

It's reasonable to assume that the intended interpretations are a subset 
of the graph's satisfying interpretations, since it would be pointless 
to publish a graph that is inconsistent in every interpretation from the 

>>>> - The most accurate way to understand a graph is to interpret
>>>> it in the way that the author intended it to be interpreted.
>>>> Since we have no other reliable way of knowing what that might
>>>> be, we can assume that the author's intended interpretations
>>>> for a graph are a subset of the graph's **satisfying
>>>> interpretations**.  I.e., we take the graph's meaning at face
>>>> value, rather than attempting to interpret it according to some
>>>> hidden, assumed "real world" interpretation.
>>> Yes, this is exactly what the RDF model theoretic semantics
>>> presumes. Asserting a graph effectively claims that
>>> interpretations must be such as to make it true, i.e. to satisfy
>>> it. Each graph makes some claims about how the world is
>>> structured,
>> Yes.  :)
>>> and the claims made by multiple graphs are connected by their
>>> common use of global IRIs.
>> *If* the graphs are merged.
> No. This has nothing at all to do with whether the graphs are merged
> or not. Merging (actually, because of the new treatment of shared
> blank node scopes, unioning, but let us ignore this) two graphs does
> not change the meaning of the triples in those graphs in ANY WAY.
>> If the graphs are not merged, then the RDF semantics says nothing
>> about them using the same interpretations.
> WRONG. Utterly and disastrously wrong.

*Sigh.*  No, it is not wrong.  The notion that "an IRI *always* denotes 
the same resource" is only true within a *single* application of the RDF 
Semantics to a particular graph-interpretation pair.   It is not true 
when the RDF Semantics are applied *separately* to more than one 
graph-interpretation pair.  The semantic formulas make no constraints 
across graphs that are not unioned/merged.

This may be more tedious to explain than it's worth, but I will try.

We all know that an interpretation is a function, and it's a total 
function, so there's no need to point that out again.  But an 
interpretation *only* constrains the referents of IRIs **via the 
semantic rules that are specified in the RDF Semantics**.  The mere 
*existence* of an interpretation does not constrain *anything*.  (If it 
did, then there could only be one interpretation, because an 
interpretation is a mapping.  But we know that there exists more than 
one (different) interpretation.)  The fact that an interpretation exists 
has no relevance whatsoever to the referents of any IRIs in any graph, 
or to the truth value of any graph, **unless it is applied via the 
semantic rules**.  If I *choose* to determine the truth value of a graph 
G1 WRT interpretation I1, then the RDF Semantics tells me how tto 
determine *that* truth value.  But in doing so, that does not in any way 
constrain the truth value of a *different* graph G2 (assuming G2 is not 
a subset of G1). The point is that the RDF semantics *only* says 
anything about the meaning of a graph **to which it is applied**.

This is analogous to a thermometer.  You can use it to measure the 
temperature of *any* city.  (Woohoo!)  But if you are measuring the 
temperature of Boston, that says *nothing* about the temperature of New 
York, and vice versa.  In particular, the thermometer does *not* tell 
you the temperature of two cities *simultaneously*.  You would need a 
different thermometer for that -- one with two tips.

Similarly, suppose there is an IRI U0 that appears in both G1 and G2, 
and neither graph is a subset of the other.  If you want to determine 
the truth value of graph G1 WRT interpretation I1, you can substitute G1 
and I1 throughout the semantic formulas in the RDF Semantics and you 
will see that the formulas constrain *every* occurrence of U0 in G1 to 
refer to the same resource: I1(U0). Now suppose you want to determine 
the truth value of graph G2 WRT interpretation I2.  Again, you can 
substitute G2 and I2 throughout the semantic formulas in the RDF 
Semantics and you will see that they constrain *every* occurrence of U0 
to refer to the same resource: I2(U0).  However, I1(U0) is not 
necessarily the same resource as I2(U0).  The point is that when the 
semantic rules are applied to G1 and I1, they do not at all constrain 
the truth value of G2 (assuming G2 is not a subset of G1).

Of course, one could consider the truth value of a graph GA consisting 
of the union of all possible graphs.  And in that case, the semantic 
rules would indeed constrain every occurrence of U0 to refer to the same 
resource in G1, G2 and every other graph, **with respect to determining 
the truth value of GA**.  But so what?   Nobody cares about determining 
the truth value of GA!  We all know that its truth value would be false 
anyway.  People care about the truth values of *specific* graphs of 
their choosing.

The notion that "an IRI *always* denotes the same resource" only applies 
to a *single* application of the RDF Semantics to a particular 
graph-interpretation pair.

The problem here is that the statement "an IRI always denotes the same 
resource" is inherently meaningless without additional assumptions about 
the context in which it is intended.  *You* are assuming that the 
context is from the perspective of *inside* a single application of the 
RDF Semantics.  In contrast, I am assuming an external perspective, in 
which a user may *choose* to apply the RDF Semantics to different 
graph-interpretation pairs, and then look at how the semantic formulas 
constrain those graph-interpretation pairs.  This perspective is 
relevant because different applications apply different interpretations 
to different graphs.  Hence, it is natural that the truth value of 
different graphs would be determined WRT different interpretations, and 
that may cause problems when those graphs are unioned, because in the 
graph that is the union, the RDF Semantics assumes that an IRI always 
denotes the same resource.

> The RDF semantics speaks of
> interpretations, and each interpretation fixes the referent of every
> IRI (and the IEXT mappings, ie what the relational extension is when
> this thing is understood to be a relation), and hence the truthvalue
> of every triple. Every *possible* triple, even ones that have not yet
> ever been written down.

No, an interpretation does *not* fix the referent of every IRI.  An 
interpretation is simply a *function*.  That function *can* be applied 
to any argument.  But that function *only* constrains the referents of 
things **to which it is applied in the semantic rules**.  If a function 
is not applied in the semantic rules then it is irrelevant.  It *only* 
becomes relevant **to the extent that it is used in the semantic 
rules**.  And that means that if it is applied to graph G1 in one of the 
semantic rules, then it constrains G1 -- it does *not* constrain G2 
unless G2 happens to be a subset of G1.

> The truth conditions defined by the semantics
> have got nothing whatever to do with what graphs have been published
> or created or merged on the Web.

Agreed.  The Web is irrelevant to the semantics.  But the truth 
conditions have *everything* to do with the choice of graph and 
interpretation to which they are applied, and the user is free to choose 

>> Of course, any two graphs *may* be merged.  But the point is that
>> it is sometimes useful to look at the semantics of each graph
>> *individually*
> That phrase has no meaning.

Of course it has meaning!  The RDF Semantics can be applied to *any* 
graph and *any* interpretation.  I can perfectly well determine the 
truth value of graph G1 WRT interpretation I1, and I can also determine 
the truth value of a different graph G2 WRT a different interpretation 
I2.  That, in essence, is what different RDF applications do.

> The semantics does not give individual
> interpretations to a single graph, It interprets the entire
> vocabulary of all IRIs, and that in turn determines the truth values
> of triples, and that of graphs. There is no such thing as a semantics
> of "graphs individually" versus those of two graphs merged. Let G1
> and G2 be graphs and H be their merge. Then every interpretation
> assigns a truthvalue to G1 and to G2 and to H  all three of them 
> and H is true in that interpretation exactly when both G1 and G2 are
> true in it. That is exactly what the RDF semantics says about
> merging. By the way, even when H is constructed, (graphs isomorphic
> to) G1 and G2 are still subsets of it. Merging two graphs creates a
> bigger graph but it does not destroy the pre-merged graphs.
>> (if the graphs are being used separately, by separate applications)
>> just as it is also sometimes useful to look at the semantics of the
>> merge (such as if those graphs are being used together).
> The "semantics of the merge" is identical to the semantics of the two
> graphs considered separately.

No, it is *not* the same.   Again, you are assuming that there is only 
*one* interpretation.  I can determine the truth value of graph G1 WRT 
interpretation I1 and I can also determine the truth value of graph G2 
WRT interpretation I2, via two separate applications of the RDF 
Semantics.   Those are *not* necessarily the same truth values that I 
would get if I first unioned G1 and G2 to form H and then determined the 
truth values of H WRT I1 and I2.

> The RDF semantics assigns truth values
> to *all* graphs, including graphs that only have a Platonic
> existence. Consider the set of all RDF graphs that have ever been
> published on the Web at the moment I hit the period at the end of
> this sentence. Define a 'survey' graph consisting of one triple from
> each of the graphs in that set. That RDF graph, which by construction
> has never been constructed in any RDF document or datastructure, is
> one of those which has a determined truthvalue in every RDF
> interpretation.
> BTW, each triple in that graph is both in its published graph and in
> this 'survey' graph, as well as in many many others (maybe infinitely
> many, depending on how one defines "all graphs"). So what exactly
> determines how it is to be interpreted, in your view?

I'm not sure what you mean by "how it is to be interpreted", but 
assuming that you mean "how its truth value should be determined", its 
truth value should be determined by following the semantic rules in the 
RDF Semantics spec, for the given interpretation under which you wish to 
test its truth value.

>>>> Some benefits of looking at the formal semantics this way
>>> What "way" are you talking about? Look, *of course* each graph
>>> has a set of satisfying interpretations, and asserting the graph
>>> is saying that the world being described by the graph is one of
>>> those satisfying interpretations. (Or if we want to give authors
>>> the ability to be vague about exactly what they are talking
>>> about, then the interpretations of whatever the author had in
>>> mind are a subset of the satisfying interpretations.) And of
>>> course we should take a graph at face value, as you put it, as
>>> saying exactly this. All this is *exactly* what the current
>>> semantics itself says (or presumes). As far as I can see, you are
>>> simply agreeing with standard model-theoretic intuitions here.
>> I am *entirely* agreeing with the model-theoretic semantics.  But
>> apparently, as has been evidenced in this discussion, our
>> intuitions sometimes differ.
> It not a matter of intuitions. If you agree with the RDF model
> theory, then you agree that IRI reference is global (because that is
> what the model theory says.) But you do not agree with the latter, so
> you apparently do not agree with the former. BTW, your comment ot the
> RDF WG illustrates this perfectly. The RDF specs say that IRIs have a
> global meaning. You objected to this. So apparently you do *not*
> agree with the current RDF model. Or do you? You can't have it both
> ways.

I agree with the formal semantics, but the formal semantics does not 
(and cannot) constrain the choice of graphs and interpretations to which 
the semantics are applied.  And in practice, different interpretations 
are used for different purposes/applications and are applied to 
different graphs.

>>>> Is this making any more sense to you?
>>> No. I don't know what the "it", that is supposed to provide all
>>> these advantages, actually is. If it is the idea that asserting a
>>> graph amounts to saying that the intended interpretation is one
>>> of those satisfying the graph, then this is what model theory
>>> says already. If it is the idea that an IRI can refer to one
>>> thing in one graph and a different thing in a different graph,
>>> then that is false (by definition)
>> Wrong.  Stop right there.  Please re-read that last sentence, and
>> notice that it did *not* stipulate a particular interpretation.
>> I.e., it was *not* scoped to a single interpretation.
> It does not matter. In fact, it is so scoped, implicitly, simply by
> its referring to what an IRI refers to. (It did not say, what an
> occurrence of an IRI in a graph refers to.) An interpretation, you
> may recall, is simply a mapping which identifies one way to interpret
> the universe of IRIs. But all those 'ways to interpret'  ALL of them
>  treat all occurrences of an IRI in the same way. So none of them
> treat different occurrences of a single IRI differently when they
> occur in different graphs. This is true if we are talking about one
> interpretation or about several interpretations or about all
> interpretations. There is simply no construct that is defined, or
> AFAIK can possibly be defined, within the framework of the current
> RDF semantics, which allows two occurrences of a given IRI to be
> assigned distinct referents, under any circumstances whatever. (Just
> for the record, you have not yet defined your notion of "refer in a
> graph" with enough precision to analyse it. I do not believe that you
> will ever be able to do so, but would be most interested to be proved
> wrong.)

It isn't done with a *single* application of the semantics.  It is done 
by applying the semantics more than once, separately, to different 
interpretation/graph pairs.

>> If you had instead said: "the idea that, **within a particular
>> interpretation**, an IRI can refer to one thing in one graph and a
>> different thing in a different graph, then that is false", then
>> that would be perfectly true.
> An interpretation is simply a formal way of expressing what it is
> that IRIs refer to. IRIs refer, RDF assumes, globally: that is, the
> IRI itself does the referring, and all tokens of that IRI refer in
> the same way. I agree this may be an idealization, but it is a widely
> assumed presumption, and (which is the point here) the RDF
> specifications require, normatively, that IRIs be interpreted in this
> way when considering RDF entailment.
> Even if one wants to argue (as many have done, including myself on
> occasion) that this idealization is unrealistic, and that we need to
> have a way for different occurrences of a single IRI to denote
> differently in context-sensitive ways, the idea of graphs being the
> required contexts does not make sense. A given *occurrence* of an IRI
> in a single triple is simultaneously "in" an infinitude of different
> graphs, so the idea of *the* containing graph is meaningless.
> Something else has to provide the needed context.   And the truth
> conditions have to be rewritten so as to include the context
> (whatever that context is presumed to be) into the semantic
> equations. Of course, this can all be done (in a variety of ways) but
> the result is a different semantics from that described in the RDF
> specs.
>> But the whole point of my long explanation above about a graph's
>> intended interpretations is that is is perfectly normal and natural
>> to think of applying different interpretations to different
>> graphs.
> And I have already explained, many times, why this is neither normal
> nor correct, as far as the RDF specs are concerned. In fact it is
> explicitly prohibited when considering validity of RDF inference.

Nothing in the specs prohibits, nor can it possibly prohibit, one from 
determining the truth value of I1(G1) and I2(G2), given interpretations 
I1 and I1 and graphs G1 and G2.  These are two separate applications of 
the semantic rules.

When a function f of two arguments is defined, nothing prevents that 
function from being applied to two different pairs of arguments.  The 
formal semantic rules are, in essence, parameterized by graphs and 
interpretations.  They can be applied to any graph/interpretation pair. 
  There is no way to prevent them from being applied to two different 
graph/interpretation pairs.  The result, of course, is two *separate* 
semantic analyses.  This corresponds more closely to the real world use 
of IRIs and graphs than by assuming that the same interpretation will be 
applied to every graph.

This does *not* capture the semantics of applying two different 
interpretations to different graph within a *single* application of the 
semantic rules.  To do that would require extending the semantic rules 
to add the notion of context, as you have pointed out several times. The 
consequence of this is that there are no semantic rules, for example, 
for *combining* two graph-interpretation pairs in order to consider the 
union of those graphs.  This means that users are out of luck if they 
want to combine graphs that have different interpretations.  But that's 
okay, because that reflects the fact of life: it isn't easy to usefully 
combine two graphs that are true under different interpretations. 
(Doing so requires first transforming one or both graphs to split each 
IRI whose referent differs between the two interpretations.)

>> Thus, when a single interpretation has not stipulated, it is
>> *wrong* to assume that a single interpretation is being applied.
> The text to which you are objecting does not assume this. The point
> at issue between us applies in *every* interpretation.

The text is this: "IRIs have global scope: Two different appearances of 
an IRI denote the same resource."

If different appearances of an IRI are considered in different 
interpretations, then they might not denote the same resource in those 
interpretations.  Here is a simple example:

This is, in fact, what happens in real life when different graphs are 
used for different purposes.

>> I accept the fact that your intuition may be different than mine --
>> and perhaps everyone else's also.  But you *cannot* make
>> unqualified statements like that, containing free variables, and
>> expect everyone else to have the same intuition about what they
>> mean as you do.  If you want to make a statement that is only true
>> in one interpretation, then you *must* stipulate that single
>> interpretation.
> David, let me try to respond to you in the manner which your tone
> here seems to invite. You may go to hell. You do not have the right
> nor the intellectual authority to tell me what I must or must not do
> or say. (And your point here about 'free variables" is weird enough
> to be something out of Monty Python, by the way.)  You are just
> WRONG. I have explained to you that you are wrong, how you are wrong,
> and given you actual mathematical proofs that you are wrong.
> Apparently none of this has made the slightest impact on your
> misapprehensions. So I fail to see that prolonging this conversation
> is likely to produce any more enlightenment or to be any use to
> either of us.

I'm sorry this has been such a difficult and painstaking discussion, and 
I certainly cannot force you to understand what I'm saying.  But even if 
you have not grokked what I'm saying, I know that at least some others 
have understood at least some of what I've said.  So perhaps others can 
find better ways of explaining my points than I have found.

> It is not a matter of rival intuitions.  I understand your intuition,
> but you are using the wrong formal way to describe it. Your 'refer in
> a graph' idea simply does not make sense,

Of course it makes sense!  That's exactly what happens in real world use 
of RDF!

> and if you were to fix it
> so that it does make sense, you would have a different RDF semantics.
> None of this is remotely a matter for debate or opinion.
> You think that you have a rival intuition which makes a different
> kind of sense of conventional model theory. In which case, let me
> urge you to write up your account of this rival intuition in enough
> detail and with enough precision that it can be published in one of
> the many venues for such reviewed publications, and publish it there
> so that everyone can understand it and refer to it.

You are *badly* misunderstanding what I'm saying.  Writing such a paper 
would make no more sense than it would for me to write a paper pointing 
out that, if someone defines a function f that can take any integer 
argument, that function can be separately applied to two *different* 
arguments, both f(5) and f(10).  Duh.

> And until you do
> this, I suggest that you might consider refraining from commenting
> further on the matter in public, as all that you are succeeding in
> doing by such comments is convincing a wider and wider set of readers
> that you do not know what you are talking about.

Thanks, but no thanks.  If people conclude that I don't know what I'm 
talking about then so be it.

>> So, returning to what "it" is, a central theme here is the idea
>> that different graphs have different sets of intended
>> interpretations.
> Of course they might. Nobody has ever disputed this. But the
> *intended* interpretations refer only the mental state of the author
> of the RDF, and that is a matter that will always be inaccessible to
> objective description. The RDF semantics is concerned only with what
> the published graphs actually say, not what they were intended to
> say.

I addressed this above already, so I'll avoid repeating myself.

>> Therefore, it is entirely normal and sensible to think and talk
>> about the same IRI denoting different things in different graphs
> I will not argue about normality or sensibleness, but I will continue
> to assert that the actual published normative RDF specs are based on
> the assumption that IRIs denote independently of the graphs they
> occur in: put another way, that it is IRIs that denote, not
> occurrences of IRIs in graphs.

No, the formulas are written WRT terms and graphs.  As explained above, 
the mere existence of an interpretation function constrains nothing.  It 
only constrains something when it is applied in the semantic formulas, 
and those are written WRT terms and graphs.

>> : an IRI may denote one thing in graph A's intended
>> interpretations, and a different thing in graph B's intended
>> interpretations.
>> Notice that if I were to commit the same sin that you committed
>> above (in making a statement with unbound variables) I would
>> shorten that last sentence into: An IRI may denote one thing in
>> graph A and a different thing in graph B.  And the problem seems to
>> be that when you read "An IRI may denote one thing in graph A and a
>> different thing in graph B" *your* intuition causes you to read
>> that sentence as "**Within a given interpretation**, an IRI may
>> denote one thing in graph A and a different thing in graph B", and
>> the sentence is obviously false by that reading.  But *my*
>> intuition causes me to read it as "An IRI may denote one thing in
>> graph A's **intended interpretations** and a different thing in
>> graph B's *intended interpretations**", and the sentence is
>> obviously true by that reading.
> It may be true with that reading (which has never surfaced in these
> conversations before this email, by the way) but that is irrelevant
> to how RDF is used on the Web and to RDF semantics, as I have
> explained above.

I do not understand why you say that that reading is not relevant to how 
RDF is used on the Web.  Can you explain?  AFAICT, that is *exactly* how 
people actually use RDF.  We *want* them to always use a given IRI to 
denote exactly the same thing, but they *don't*.  They in fact have 
*different* interpretations in mind when they write their RDF graphs, 
and they apply *different* interpretations when they consume those 
graphs.  And, according to the RDF Semantics, those different 
interpretations sometimes map the same IRI to *different* resources.

> For the record, by the way, I do not think it always
> makes sense even with that reading (because our very thoughts might
> have multiple interpretations which allow the IRI reference to
> vary.)

I don't know exactly what you mean, but it probably isn't worth getting 
into it.   I don't think it would make sense to start talking about 
interpretations of interpretations.

>>> but in any case would not provide all these claimed advantages
>>> that 'it' is supposed to have, even if it could be made somehow
>>> true.
>> Please tell me exactly which ones you disagree with, and why.
>> Every one of them is supported by an unbroken chain of reasoning,
>> so if you're not reaching the same conclusions as me, then we need
>> to identify what differing assumptions we're making along the way
>> and why.
>>>> Have I explained myself in sufficient detail, or do you still
>>>> think that "David . . . does not properly understand the
>>>> intuitive foundations of semantics" and my points are mere
>>>> "inanity", as you previously concluded?
>>> I regret if my usage here seemed impolite, but I do (still) find
>>> your posts, including this one, to be a strange mixture of basic
>>> ideas about model theory (re)stated as though they were somehow a
>>> new insight or an alternative to the standard view (which I
>>> referred to as "inanity") and strangely stubborn basic mistakes
>>> which do, I am afraid, strongly suggest that you have not grokked
>>> the basic ideas of model theory.
>> I am kind of amazed that you *still* seem to think that I am making
>> some kind of mistake in my understanding of model theory, since it
>> seems to me that we have repeatedly confirmed that my understanding
>> corresponds *exactly* with standard model-theoretic semantics.
>> Precisely what basic ideas of model theory do you think I have not
>> grokked?
> I believe I have already answered this question in my previous
> response.
> The pattern of our conversation is like this: A entails B. You
> repeatedly assert that you agree with and accept A, but you refuse to
> agree with B. I conclude that, in spite of your protestations, you do
> not really understand and agree with A, since if you did, you would
> see why B follows immediately from it.

Actually, I think the pattern is more like this:
Pat: "An IRI denotes the same resource in *every* graph!"

David: "Not if those graphs are evaluated under different interpretations."

Pat: "But that doesn't make sense!"

David: "Sure it does.  The semantics can be applied to *any* 
interpretation/graph pairs.  The semantics can be used to separately 
evaluate both I1(G1) and I2(G2)."

Pat: "But you don't understand model theory!"

David: "What part?"

Pat: "An interpretation is a total function!  It applies to *every* 
graph!  Therefore an IRI denotes the same resource in *every* graph!"

David: But the semantics can determine the truth value of *any* 
graph-interpretation pair, right?

Pat: Right, but . . .

(Repeat from the beginning.)

>>>> And do you *still* think I merely need to go read a book on
>>>> model theory, or have we now (I hope) got past that?  If not,
>>>> what aspects of model theory do you still think I
>>>> misunderstand?
>>> Well, I guess, the basic idea of an interpretation. An RDF
>>> interpretation, by definition, is a mapping from IRIs to
>>> referents. It is not a mapping from IRIs-in-graphs or from
>>> IRI-occurrences or from IRIs-in-a-context. Ergo, every
>>> interpretation treats all occurrences of an IRI in the same way,
>>> as referring to the same thing, regardless of which graphs the
>>> IRI happens to occur in.
>> Yes.  I don't know what I have to do to convince you that I
>> understand that, but believe me, I do.
> You apparently don't understand the consequences of it. But there is
> no point in continuing this correspondence. You may do and say
> whatever you like, and the RDF specs will say something else. I will
> not undertake to try further explanations.
>>> Therefore, the notion of what an IRI denotes "in a graph" is
>>> meaningless.
>> No, as already explained in some detail above, that does not
>> follow. The idea that an IRI denotes something "in a graph"
>> certainly *is* meaningful if you think of it in terms of the
>> graph's **intended interpretations**.  If you think of it that way,
>> it is a very sensible notion indeed.
> Perhaps, although I doubt it. But that is completely irrelevant to
> almost everything that has been said until now in all these various
> threads, which have not mentioned "intended" interpretations, but
> interpretations of graphs. The "intended" language is also completely
> absent from the RDF Semantics specs, of course.

The fact that this seems to you to be irrelevant very clearly 
illustrates how you have not grokked what I've been trying to say.  To 
be clear, the idea that different interpretations are used with 
different graphs is the *essence* of what I have been saying.


>>> This basic fact  and it is a very basic and foundational point 
>>> still seems to elude you.
>> I hope the above explanations have helped you to see why I keep
>> pointing out that that conclusion is wrong.
>>> To emphasize, this is not a "perspective" which admits
>>> alternatives, it is simply a fact about how interpretations are
>>> defined.
>> The fact that an interpretation is (by definition) a mapping from
>> IRIs to referents has not eluded me at all.  I am fully aware of
>> that.  The perspective part comes in after that, when talking about
>> a graph's intended interpretations.  One perspective is to assume a
>> single (real-world) interpretation to by which all graphs are
>> evaluated. Another perspective is to evaluate each graph by its
>> intended interpretations.  Both views are fully consistent with
>> standard model theory, but AFAICT they seem to appeal to different
>> intuitions.
> The only useful way to proceed is to consider all graphs that come
> your way, and draw valid conclusions from them. This amounts to
> giving RDF free rein to interpret its graphs in any way that the
> semantics permits. Neither of your "perspectives" is relevant to the
> description of  RDF entailment.
>>>> The bottom line here is that some of the statements -- and
>>>> intuition -- in the existing RDF drafts are just plain *wrong*
>>>> and need to be corrected.  In particular, the statement in RDF
>>>> Concepts that says "IRIs have global scope: Two different
>>>> appearances of an IRI denote the same resource" is just
>>>> factually *wrong*.
>>> It is a presumption of the RDF data model.  The semantics, in
>>> particular, is based on it. I don't quite see how it can be
>>> factually wrong, since RDF *defines* the notion of denotation.
>>> (If it had said "identiifies" then it might be factually wrong,
>>> but it doesn't.)
>> Hopefully my explanations above have by now clarified why this is
>> wrong as stated
> For the record, they have not. If anything, your argument is now even
> more confused than it appeared to be before this email.
>> , so I won't go over it again here.  But if you disagree please let
>> me know where and why, so that I can address it.
> I already have let you know, in painful detail. I will do you the
> honor of not repeating myself yet again. As I say, if you think you
> have a case, write it up and publish it in a refereed journal.
> Good luck.
> Pat
>> Best, David
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Received on Friday, 13 December 2013 19:15:45 UTC